Preparation and properties of dinitrogen complexes of molybdenum and tungsten with trimethylphosphine as coligand : III. Synthesis and properties of cis-[W(N2)2(PMe3)4], trans-[W(C2H4)2(PMe3)4] and [M(N2)(PMe3)5] (M = Mo,W). The crystal and molecular structure of [Mo(N2)(PMe3)5]

The reduction of [WCl₄(PMe₃)₃] with dispersed sodium, under dinitrogen, gives cis-[W(N₂)₂(PMe₃)₄], while under ethylene trans-[W(C₂H₄)₂(PMe₃)₄] is obtained. The ethylene complex can also be prepared by displacement of the dinitrogen molecules in cis-[W(N₂)₂(PMe₃)₄] by ethylene at room temperature and pressure. Interaction of cis-[M(N₂)₂(PMe₃)₄] complexes (M = Mo, W), with PMe₃, under helium or argon, yields [M(N₂)(PMe₃)₅]. The molybdenum complex crystallizes in the orthorhombic space group Pnma, with a 22.063(6), b 12.106(4), c 9.745(4) Å. The Mo—P distance trans to the dinitrogen ligand (2.483(7) Å) is slightly longer than the average of the other four Mo—P bonds (2.460(5) Å).     

The formation and molecular structure of acetylcyclopentadienylsodium-tetrahydrofuranate

Acetylcyclopentadienylsodium has been isolated in crystalline form as a THF adduct from a reaction between cyclopentadienylsodium and methyl acetate in THF solution. The product has been characterized by means of a single-crystal X-ray diffraction study. {[C₅H₄CMeO]Na·THF}_n crystallizes in the monoclinic space group P2₁/c with unit cell parameters a 6.698(3), b 16.095(4), c 10.661(3) Å, β 92.93(3)° and D_c 1.17 g cm^?3 for Z = 4. Least-squares refinement led to a final R value of 0.080 based on 661 independent observed reflections. The coordination sphere around each sodium atom consists of the oxygen atoms from two C₅H₄CMeO ligands, the oxygen atom of the THF molecule, and an ion contact pair between the sodium and the five ring carbon atoms of the C₅H₄CMeO ligand.     

The charge-charge flux-overlap model for the interpretation of atomic polar tensors and infrared intensities

The interpretation of infrared intensities using the concepts of atomic polar tensors divided according to contributions from charge-charge flux-overlap (CCFO) terms appears in general to be quite useful. Here we shall illustrate this analysis for the F and H atoms in chemically related molecules to show how their properties vary for CF, CH, OF and OH bonds as R varies in RCF, RCH and ROH molecules.     

Sharp Morawetz estimates

We prove sharp Morawetz estimates — global in time with a singular weight in the spatial variables — for the linear wave, Klein-Gordon, and Schrödinger equations, for which we can characterise the maximisers. We also prove refined inequalities with respect to the angular integrability.     

Irregularities of distribution with respect to strips

Summary A lower bound is established for the strip discrepancy of a broad class of point distributions. The bound implies unbounded strip discrepancy for equally weighted point distributions under favorable conditions. The methods of proof use notions from integral geometry.     

Optimal and robust contracts for a risk-constrained principal

The theory of risk measurement has been extensively developed over the past ten years or so, but there has been comparatively little effort devoted to using this theory to inform portfolio choice. One theme of this paper is to study how an investor in a conventional log-Brownian market would invest to optimize expected utility of terminal wealth, when subjected to a bound on his risk, as measured by a coherent law-invariant risk measure. Results of Kusuoka lead to remarkably complete expressions for the solution to this problem. The second theme of the paper is to discuss how one would actually manage (not just measure) risk. We study a principal/agent problem, where the principal is required to satisfy some risk constraint. The principal proposes a compensation package to the agent, who then optimises selfishly ignoring the risk constraint. The principal can pick a compensation package that induces the agent to select the principal’s optimal choice.     

Indecomposable continua and the Julia sets of polynomials, II

We find necessary and sufficient conditions for the connected Julia set of a polynomial of degree d?2 to be an indecomposable continuum. One necessary and sufficient condition is that the impression of some prime end (external ray) of the unbounded complementary domain of the Julia set J has nonempty interior in J. Another is that every prime end has as its impression the entire Julia set. The latter answers a question posed in 1993 by the second two authors.We show by example that, contrary to the case for a polynomial Julia set, the image of an indecomposable subcontinuum of the Julia set of a rational function need not be indecomposable.     

Sharp van der Corput estimates and minimal divided differences

We find the sharp constant in a sublevel set estimate which arises in connection with van der Corput's lemma. In order to do this, we find the nodes that minimise divided differences. We go on to find the sharp constant in the first instance of the van der Corput lemma. With these bounds we improve the constant in the general van der Corput lemma, so that it is asymptotically sharp.